Question

If two parallel lines are intersected by a transversal line, then the bisectors of the interior angles forms a:

• A. square
• B. rectangle
• C. parallelogram
• D. trapezium

Answer 1

Answer: (B)

rectangle

If two parallel lines are intersected by a transversal line, then the bisectors of the interior angles forms a rectangle.

To prove this, we will prove that the angle between two adjacent bisectors is $90°$.

Let's start with bisectors of $\angle BAC$ and $\angle BAE$. We know that the sum of these two angles is $180°$.
Then the angle between the bisectors of these angles is
$\frac{\angle BAC}{2}+\frac{\angle BAE}{2}=\frac{(\angle BAC+\angle BAE)}{2}=\frac{180°}{2}=90°$.
Similarly, we can prove that the angle between the bisectors of $\angle ABC$ and $\angle ABD$ is $90°$.

Now, let the bisectors of $\angle BAC$ and $\angle ABC$ intersect at point $G$.
We know that the sum of angles of a triangle is $180°$. Using this fact,
$\angle AGB=180°-\frac{(\angle BAF+\angle ABC)}{2}$
$\angle AGB=180°-\frac{180°}{2}=180°-90°=90°$
Similarly, we can prove that the angle between bisectors of $\angle BAE$ and $\angle ABD$ is $90°$.
Since, all the angles formed by the bisectors of the interior angles are $90°$, the bisectors of the interior angles forms a rectangle.